A glass bottle filled with oil weighs $590$ grams. After Sophia uses $200$ milliliters of oil, the bottle of oil weighs $400$ grams. The weight $M$ of the bottle of oil, in grams, is a function of $V$, the volume, in milliliters, of oil Sophia has used. Write the function's formula. $M=$
Explanation: The density of oil is constant, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $M= mV+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that the bottle of oil initially weighs $590$ grams, so the $y$ -intercept ${b}$ is ${590}$, and our function looks like $M={m}V+{590}$. We also know that after Sophia uses $200$ milliliters of oil, the bottle of oil weighs $400$ grams, which means when $V=200$, $M=400$. We can use this and the $y$ -intercept to find ${m}$ : $\begin{aligned} {m}&=\dfrac{M_2-M_1}{V_2-V_1} \\\\ &=\dfrac{400-590}{200-0} \\\\ &=\dfrac{-190}{200} \\\\ &=-\dfrac{19\cdot\cancel{10}}{20\cdot\cancel{10}} \\\\ &={-0.95} \end{aligned}$ This means for each milliliter of oil Sophia uses, the weight of the bottle of oil decreases by $0.95$ grams. Since ${m}={-0.95}$ and ${b}={590}$, the desired formula is: $M={-0.95} V + {590}$